Applications of algebraic manipulation programs in physics

Abstract

Barton andJ P Fitch algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review.This review was completed in December 1971.

Constructing Galerkin's approximations of invariant tori using MACSYMA

“…Jefferys [25] has pointed out that workers in celestial mechanics, constructing, for example, the theory of the orbit of a celestial body, have had to deal with expansions involving hundreds or even thousands of terms. Barton and Fitch [3] have noted that even in simple problems involving the manipulation of functions, problems starting from and resulting in quite short expressions frequently lead to intermediate expressions of inordinate length in the course of computation. They refer to this as the problem of 'blow-up'.…”

“…As an example of this form of representation consider the simple expression (3x2y + x)sin2(v + 2)cos2(2v + u) (31) and note that it is not in the Poisson series form. However, it can be rewritten in the form 3 1 + xZy + x (32) which, although seemingly more complex than (31), can be represented computationally very compactly in a list structure.…”

“…There were four solutions for (57), given by 1. a(~ ) = a~ ) =0, 2. a~ ) = 0~ a~)=4, 3. a~ ) = 4, a(~ )=0,…”

Experiments with a symbolic programming system for complex analysis

“…There is a good survey of the scope of SC in the review by Barton and Fitch [1], even though this survey is over four years old; since 1972, much of the work in the field has gone into finding faster or more compact algorithms for operations which were already available at that time in the best systems. Where there are isolated techniques of interest in applied mathematics which have not been implemented to any great extent in SC systems, the reason is usually to be traced to general theoret.ieal difficulties.…”

Problem #4

“…BX, Then using the inverse transform one obtains x. and X. as "functions of t and the problem is solved) I We shall not attempt to produce the full mathematical derivation of the Lie transform here but refer the interested reader to Deprit (1969), Mersman (1970) or Barton and Fitch (1972). Consider instead the simple example of the harmonic oscillator in one space variable x whose Hamiltonian is…”